Resumen
Este artículo presenta un algoritmo nuevo basado en una extensión del método algorítmico simplex de Nelder Mead para la identificación de al menos un óptimo local cuando se usa en problemas no lineales enteros mixtos irrestrictos. Este método algorítmico, denominado Algoritmo Simplex Entero Mixto (ASEM) por el autor, se basa en una doble estructura de símpleces que está compuesta por una estructura simplex real de dimensión n (simplex real) y otra estructura simplex entera de igual dimensión n (simplex entero). Las operaciones propuestas originalmente por Nelder Mead se ejecutan sobre el simplex real. Por otra parte, un grupo de nuevas operaciones presentadas en este artículo se emplean en el simplex entero. Este conjunto de nuevas operaciones junto con las operaciones originales del método de Nelder Mead garantizan que en cada iteración del ASEM se arroje un nuevo punto de prueba definido en el campo numérico de dimensión 2n enteros mixtos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
, con el objetivo de garantizar la identificación del óptimo local entero mixto, sin convertir las variables enteras en variables reales.
Abstract
This article presents a new algorithm based on the Nelder-Mead simplex algorithmic method for identifying a local optimum, at least on unconstrained nonlinear mixed-integer problems. The algorithmic method, Integer Mixed Simplex Algorithm (IMSA), so called by the author, is based on a double simplex structure, which is composed of a real n -dimensional simplex structure (real simplex) and an integer n -dimensional simplex structure (integer simplex). The original Nelder-Mead operations are applied on the real simplex. Meanwhile, a novel group of operations are applied on the integer simplex. This new set of operations, together with the original Nelder-Mead operations, guarantee a new trail point at each IMSA iteration in the search of the local optimum in the integer real mixed 2n -dimensional numerical field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
without the need of integer to real conversions.
Palabras clave
Método simplex ; Método de Nelder Mead ; optimización entera mixta ; búsqueda directa
Keywords
Simplex method ; Nelder-Mead method ; Mixed-integer optimization ; Direct search
1. Introducción
El desarrollo de métodos de optimización para su empleo en la búsqueda de soluciones a problemas enteros mixtos ocupa un lugar importante en diversos campos de la ingeniería, debido a las amplias posibilidades de aplicación en problemas de diseño óptimo, cuando en este tipo de problemas las variables enteras no pueden ser relajadas a variables reales, que suelen ser abordados con el propósito de emplear algún método algorítmico de optimización desarrollado para variables reales. Además, esta situación de inconvertibilidad de las variables enteras a variables reales se presenta con frecuencia cuando la función objetivo del problema de optimización es representada mediante la función de desempeño del sistema bajo estudio, la cual es medida mediante algún modelo de simulación que depende tanto de variables reales como de variables enteras. Bajo estas circunstancias es inconveniente transformar el problema de optimización entero mixto (OEM) a un problema de optimización de variables reales, debido al hecho de que existen variables de decisión que no pueden ser tratadas como variables reales. Como ejemplo de sistemas que son definidos por variables enteras mixtas se tienen: el número de cajeros en un supermercado y su respectiva tasa de atención; la capacidad de clientes que pueden aguardar a ser atendidos en cada cola correspondiente a su taquilla de una agencia bancaria y la tasa de servicio de cada cajero; el número de dientes de cada engranaje que conforman una caja reductora y la dimensión de cada engranaje; el número de etapas de un sistema amplificador de señales y la ganancia de cada etapa de amplificación entre otros.
Es en esta situación donde los métodos de búsqueda directa, llamados así por primera vez por Hooke y Jeeves [1] , pueden brindar una solución a este tipo de problemas de optimización, bien sea por no disponer de una expresión matemática de la función objetivo o por no contar con el gradiente de esta, principalmente cuando se requiere optimizar un proceso que depende tanto de variables reales como de variables enteras, lo que exige la aplicación de estrategias de exploración que permitan avanzar a un mejor punto de prueba en cada iteración, con el fin de identificar la solución óptima del problema.
En este artículo se presenta una extensión del método simplex de Nelder Mead (MSNM) [2] a problemas OEM cuando la función objetivo es no lineal y no está sujeta a restricciones. El Algoritmo Simplex Entero Mixto (ASEM), denominado así por el autor debido a que se basa en el patrón de búsqueda conocido como simplex, se fundamenta en las operaciones definidas por el MSNM y en un conjunto de nuevas operaciones propuestas por Brea [3] , que son ejecutadas únicamente sobre las variables enteras.
De acuerdo con Brandts et al. [4] , puede decirse que un simplex (d -simplex) definido en el espacio euclidiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^d}
, siendo d un entero positivo, es un casco o envoltorio convexo de d + 1 puntos, los cuales no todos pertenecen al mismo hiperplano definido en el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^d}
, y cuyos puntos son denominados vértices del simplex.
En consecuencia, un 1-simplex es un segmento de recta en el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^1}
; un 2-simplex es un triángulo en el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^2}
; un 3-simplex es un tetraedro en el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^3}
, y así sucesivamente.
Debido a que los problemas de optimización enteros mixtos son abordados en esta investigación mediante estructuras de simplex, se hace necesario definir 2 símpleces: uno para las variables reales, denominada aquí simplex real y otro que es empleado en el campo entero, denominado en este artículo simplex entero. Sobre estos símpleces es definido un conjunto de operaciones a fin de explorar en el espacio de búsqueda entero mixto y así poder identificar la solución óptima local del problema.
El ASEM ha mostrado ser lo suficientemente eficiente a problemas OEM irrestrictos [3] . No obstante, también ha sido probado con problemas no lineales enteros mixtos bajo restricciones a través de funciones de penalización, lo cual ha permitido comprobar que el ASEM es lo suficientemente eficaz en la identificación de óptimos locales en problemas de OEM bajo restricciones.
Mediante la ejecución de un número importante de ejemplos numéricos, Brea [5] ha comprobado el buen desempeño del MSNM cuando la función objetivo está sujeta a ruido, y además cuando el problema está sujeto a restricciones lineales. Sin embargo, existen condiciones particulares en donde el método algorítmico que él desarrolló [5] converge muy lentamente al óptimo.
Por otra parte, Barton e Ivey [6] y Humphrey y Wilson [7] han estudiado los parámetros que ofrecen el mejor desempeño del MSNM, cuando la función objetivo de un problema de optimización irrestricto está influenciada por ruido.
Estos hechos han motivado al autor a desarrollar un método algorítmico basado en el MSNM en el campo de los números enteros mixtos.
El resto de este artículo está estructurado como sigue: en la sección 2 se describe matemáticamente el tipo de problema de OEM que hay que resolver mediante el ASEM; en la sección 3 se enuncian un conjunto de definiciones preliminares que permitirán exponer claramente las proposiciones que definen las nuevas operaciones que se deben emplear sobre el simplex entero, además de facilitar la explicación del algoritmo; en la sección 4 se enuncian las proposiciones que definen las nuevas operaciones que son aplicadas sobre el simplex entero; en la sección 5 son establecidos los procedimientos empleados en el ASEM; en la sección 6 se ofrece una explicación del ASEM mediante una inédita representación basada en grafo; en la sección 7 se presenta un modo de flexibilizar el ASEM con el propósito de emplearlo en la optimización de problemas, en donde el número de componentes reales es diferente al número de componentes enteras de las variables de decisión; la sintonización o los ajustes de los parámetros del ASEM se muestran en la sección 8 ; en la sección 9 se muestra un conjunto de experimentos numéricos con el objeto de mostrar la eficacia del ASEM; una comparación del ASEM con otros algoritmos de optimización en el problema de diseño óptimo de un tanque presurizado se presenta en la sección 10 ; finalmente, se presentan conclusiones y propuestas de futuras investigaciones en la sección 11 .
2. El problema
Problema 1.
Sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{x}}\in {\mathbb{R}}^n}
el conjunto de n variables reales independientes e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}\in {\mathbb{Z}}^m}
el conjunto de m variables enteras independientes. Entonces, el problema de minimización irrestricto entero mixto puede ser expresado por:
|
( 1) |
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\boldsymbol{\mbox{x}},\boldsymbol{\mbox{y}}):{\mathbb{R}}^n\times {\mathbb{Z}}^m\rightarrow \mathbb{R}}
es una función objetivo no lineal.
3. Definiciones preliminares
En esta sección se establecerá un conjunto de definiciones que facilitarán la explicación del método algorítmico desarrollado.
Definición 1 Espacio euclidiano entero mixto.
Se entenderá por espacio euclidiano entero mixto de dimensión n + m el conjunto de puntos definidos por el producto cartesiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^m}
, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}}
es el conjunto de los números reales y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{Z}}
es el conjunto de los números enteros.
Es importante señalar que, debido a la naturaleza del problema, el espacio de exploración donde deben ejecutarse las operaciones matemáticas es el espacio euclidiano entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^m}
, cuya dimensión corresponde a n + m , indicando respectivamente así el número de variables reales y enteras que conforman el problema.
Definición 2 Vector precedente.
Sea una función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\boldsymbol{\mbox{w}}):W\rightarrow \mathbb{R}}
y sean a y b 2 vectores cualesquiera definidos en un dominio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle W}
. Se dice que a precede o es igualmente precedente a b , y se denota mediante a ⪯ b , si f (a ) ≤ f (b ).
La definición 2 también se puede entender en el sentido estricto, esto es: un vector precede estrictamente a otro si la relación entre los valores de la función es estrictamente menor, es decir, a ≺ b si f (a ) < f (b ).
Definición 3 Vector subsecuente.
Sea una función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\boldsymbol{\mbox{w}}):W\rightarrow \mathbb{R}}
y sean a y b 2 vectores cualesquiera definidos en un dominio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle W}
. Se dice que a es subsecuente o es igualmente subsecuente a b , y se denota como a ⪰ b , si f (a ) ≥ f (b ).
También la definición 3 se puede entender en su sentido estrictamente subsecuente. En este caso se dice que el vector a es estrictamente subsecuente al vector b , y se denota como a ≻ b , si f (a ) > f (b ).
Definición 4 Vector entero mixto.
Sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{x}}\in {\mathbb{R}}^n}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}\in {\mathbb{Z}}^m}
2 subvectores que conforman un vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{v}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left[\boldsymbol{\mbox{x}}^t\quad \boldsymbol{\mbox{y}}^t\right]}^t
. Se dice que el vector v es un vector entero mixto en el espacio euclidiano entero mixto de dimensión n + m , para indicar respectivamente la dimensión de las componentes reales y enteras, si el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{v}}\in {\mathbb{R}}^n\times {\mathbb{Z}}^m}
.
Definición 5 Subvector real.
Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{v}}\in {\mathbb{R}}^n\times {\mathbb{Z}}^m}
un vector entero mixto. Se dice entonces que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{x}}\in {\mathbb{R}}^n}
es el subvector real de dimensión n , si sus componentes representan las cantidades reales del vector entero mixto v .
Definición 6 Subvector entero.
Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{v}}\in {\mathbb{R}}^n\times {\mathbb{Z}}^m}
un vector entero mixto. Se dice que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}\in {\mathbb{Z}}^m}
es el subvector entero de dimensión m , si sus componentes representan cantidades enteras del vector entero mixto v .
Debido a que el método algorítmico desarrollado para resolver problemas enteros mixtos requiere que las operaciones sobre cada subvector se ejecuten de forma independiente, es decir, que los nuevos puntos sean definidos por operaciones sobre los subvectores reales y sobre los subvectores enteros de forma separada, y que se permita así la coexistencia de ambos conjuntos de subvectores sin que ellos cambien sus condiciones en cuanto al espacio numérico que lo definen, se hacen necesarias las definiciones de operaciones sobre los subvectores pertenecientes a un mismo campo numérico.
Como consecuencia de las exigencias impuestas, se estudiará el caso particular donde n = m , quedando entonces las siguientes definiciones en el espacio euclidiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
. No obstante, se propondrá una adecuación a los problemas en el espacio euclidiano entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^m}
para cuando n ≠ m , y de esta forma poder aplicar el algoritmo desarrollado a situaciones en donde el número de variables reales y enteras son distintos.
Definición 7 Centroide.
Sea un conjunto de k puntos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{p}}_i=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left(\boldsymbol{\mbox{x}}_i^t\quad \boldsymbol{\mbox{y}}_i^t\right)}^t
definidos en el espacio euclidiano entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
para cada i = 1, … k . Se dice que el centroide Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{p}}}}
es el centro de masa de los k puntos, si a cada uno de los puntos se le ha considerado una masa mi . En el caso de que la masa de cada uno de los puntos pi sea igual, el centroide es determinado por
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( 2) |
Observación 1.
Nótese que el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{p}}}}
no necesariamente pertenece al espacio euclidiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
. Más aún, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{p}}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left({\overline{\boldsymbol{\mbox{x}}}}^t,{\overline{\boldsymbol{\mbox{y}}}}^t\right)}^t
, donde
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( 3) |
Definición 8 Simplex entero mixto.
Se dice que un q -ésimo simplex entero mixto en el espacio euclidiano entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
es un conjunto de diferentes puntos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{v}}_i=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left[\boldsymbol{\mbox{x}}_i^t\quad \boldsymbol{\mbox{y}}_i^t\right]}^t
para todo i = 1, …, ν , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{x}}_i\in {\mathbb{R}}^n}
corresponde al i -ésimo subvector real de dimensión n , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_i\in {\mathbb{Z}}^n}
denota cada i -ésimo subvector entero de dimensión n , ν = n + 1 representa el número de vértices del simplex entero mixto, los cuales no todos pertenecen a la misma hipercara definida en el espacio euclidiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
, y cada uno es representado por un vector de dimensión n + n . El q -ésimo simplex entero mixto puede entonces ser representado en notación matricial como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [\boldsymbol{\mbox{v}}_1^{\left[q\right]}\quad ;\boldsymbol{\mbox{v}}_2^{\left[q\right]}\quad ;\cdots \quad ;\boldsymbol{\mbox{v}}_{\nu }^{\left[q\right]}]
.
De acuerdo con la definición 8 , se establece que
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( 4) |
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{X}}_{\nu }^{\left[q\right]}\in {\mathbb{R}}^{n\times (n+1)}}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}\in {\mathbb{Z}}^{n\times (n+1)}}
representan respectivamente el simplex real y el simplex entero.
Un concepto importante que debe conocerse es el referente al significado de simplex colapsado o degenerado, el cual se identifica cuando todos los vértices del simplex pertenecen a un mismo hiperplano.
Definición 9 Menor simplex entero.
Sea la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [\boldsymbol{\mbox{y}}_1^{\left[q\right]}\quad ;\boldsymbol{\mbox{y}}_2^{\left[q\right]}\quad ;\cdots \quad ;\boldsymbol{\mbox{y}}_{\nu }^{\left[q\right]}]
un q -ésimo simplex entero. Se dice que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
es el más pequeño simplex entero no colapsado contenido en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^{n\times (n+1)}}
, si existe un vértice que equidista de sus otros vértices del simplex entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
en 1.
Definición 10 Aristas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
.
Una p -ésima matriz de aristas de un q -ésimo simplex entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
en el espacio euclidiano entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
se define como una matriz de dimensión 2n × n , cuya j -ésima columna representa la arista del simplex entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
entre un vértice referencial vp y vj para todo j = 1, …, ν y j ≠ p , es decir,
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( 5) |
Nótese que de acuerdo con la definición 10 , una matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{A}}_p^{\left[q\right]}}
representa el conjunto de aristas del q -ésimo simplex entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
que comparten un mismo p -ésimo vértice, es decir, son todas las aristas que se intersectan en el p -ésimo vértice del q -ésimo simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
.
Definición 11 Aristas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{X}}_{\nu }^{\left[q\right]}}
.
Una p -ésima submatriz de aristas de un q -ésimo simplex real Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{X}}_{\nu }^{\left[q\right]}}
en el espacio euclidiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n}
se define como una matriz de dimensión n × n , cuya j -ésima columna representa la arista del simplex real Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{X}}_{\nu }^{\left[q\right]}}
entre un vértice referencial xp y xj para todo j = 1, …, ν y j ≠ p , es decir,
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( 6) |
De igual manera puede ser definida la matriz de aristas del simplex entero, la cual se puede enunciar como:
Definición 12 Aristas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
.
Una p -ésima submatriz de aristas de un q -ésimo simplex entera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
en el espacio euclidiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
, se define como una matriz de dimensión n × n , cuya j -ésima columna representa la arista del simplex entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
entre un vértice referencial yp e yj para todo j = 1, …, ν y j ≠ p , es decir,
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( 7) |
Observación 2.
Nótese que un simplex real Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{X}}_{\nu }^{\left[q\right]}}
es colapsado si la matriz de arista Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{A}}_{x_p}^{\left[q\right]}}
es singular para todo j = 1, …, ν . De igual modo se puede afirmar que un simplex entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
es colapsado o degenerado si la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{A}}_{y_p}^{\left[q\right]}}
es singular para todo j = 1, …, ν .
Definición 13 Simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
jerarquizado.
Se dice que un q -ésimo simplex entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
es jerarquizado de acuerdo con f (v ) si v1 ⪯ v2 ⪯ ⋯ ⪯ vν , lo que significa que f (v1 ) ≤ f (v2 ) ≤ ⋯ ≤ f (vν ), en el espacio euclidiano entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^n}
.
Observación 3.
Cuando el simplex no esté jerarquizado según algún criterio, esto se indica con el símbolo tilde, es decir, como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{Y}}}}_{\nu }^{\left[q\right]}}
. Por otra parte, nótese que cuando un q -ésimo simplex mixto está jerarquizado, tanto su correspondiente simplex real Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{X}}_{\nu }^{\left[q\right]}}
como su simplex entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
están jerarquizados de acuerdo con f (v ).
Definición 14 Función signo de un entero.
Sea un número entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k\in \mathbb{Z}}
. Entonces la función signo entero de k y que es denotada por sgnd (k ) adquiere el valor de 1, si k > 0; 0, si k = 0; o −1, si k < 0.
4. Proposiciones de nuevas operaciones
Debido a que el ASEM requiere contar con un conjunto de operaciones que han de ser aplicadas sobre la submatriz de elementos enteros Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
, se requiere establecer un conjunto de proposiciones que formarán parte de los operadores que actuarán sobre el simplex entero mixto jerarquizado de acuerdo con f (v ), junto con las ya conocidas operaciones de Nelder Mead [2] , las cuales son aplicadas sobre el simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{X}}_{\nu }^{\left[q\right]}} .
Proposición 1 Punto reflejado entero.
Sea un q -ésimo simplex entero jerarquizado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
de acuerdo a f (v ), en el espacio euclidiano entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
. Si el punto reflejado entero vr a través del centroide de su hipercara Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{\nu }^{\left[q\right]}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \boldsymbol{\mbox{v}}_i\in \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}\vert \quad i= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1,\ldots ,\nu -1\rbrace
opuesta al ν -ésimo vértice vν es calculado por
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( 8) |
donde para un Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\alpha }_e\in \mathbb{N}(\geq 2)}
, el cual es el denominado coeficiente de reflexión entero, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu =}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lceil \parallel \overline{\boldsymbol{\mbox{y}}}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \boldsymbol{\mbox{y}}_{\nu }\parallel \rceil
es el valor entero próximo superior de la distancia entre yν e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{y}}}}
, dado que el operador ⌈· ⌉ aproxima el resultado a su próximo superior entero, e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{y}}}}
es determinado por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{y}}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{\nu -1}{\sum }_{i=1}^{\nu -1}\boldsymbol{\mbox{y}}_i
. Entonces, se cumple que:
- yr es vector definido en el espacio entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
;
- yr e yν están a sendos lados de la hipercara Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{\nu }^{\left[q\right]}}
.
Demostración Parte I) Denote (y1,ν , …, yn ,ν )t el vértice yν , es decir, yν = (y1,ν , …, yn ,ν )t . Entonces, de acuerdo con la ecuación (8) , se tiene que
|
donde ki ∈ { −1, 0, 1} para todo i = 1, …, n debido a la función signo discreta que es aplicada sobre cada elemento del vector. En consecuencia, cada una de las componentes de yr es entera, en virtud de que yi ,ν , αe , μ y ki son todos números enteros, y para cada i = 1, …, n su operación arroja un número entero.
Parte II) Si tanto el vértice yν como el punto yr están ubicados en sendos lados de la hipercara Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{\nu }^{\left[q\right]}}
, entonces el ángulo entre los vectores formados por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_{\nu }-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overline{\boldsymbol{\mbox{y}}}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_r-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overline{\boldsymbol{\mbox{y}}}
debe ser obtuso.
Este hecho se demuestra cuando, al realizar el producto escalar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\left(\boldsymbol{\mbox{y}}_{\nu }-\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \overline{\boldsymbol{\mbox{y}}}\right)}^t(\boldsymbol{\mbox{y}}_r- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overline{\boldsymbol{\mbox{y}}})
, este es negativo. De esta manera se tiene que de la ecuación (8) se puede escribir
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( 9) |
Entonces, al premultiplicar la ecuación (9) por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\left(\boldsymbol{\mbox{y}}_{\nu }-\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \overline{\boldsymbol{\mbox{y}}}\right)}^t
, se obtiene que
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( 10) |
Simplificando la ecuación (10) , se tiene que
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( 11) |
Si se denota a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\left(\boldsymbol{\mbox{y}}_{\nu }-\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \overline{\boldsymbol{\mbox{y}}}\right)}^t= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left(d_1,d_2,\ldots ,d_n\right)}^t
se tiene que
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( 12) |
Sustituyendo la ecuación (12) en la ecuación (11) se puede afirmar que
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( 13) |
Dado que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu =\lceil \parallel \overline{\boldsymbol{\mbox{y}}}-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \boldsymbol{\mbox{y}}_{\nu }\parallel \rceil \geq \vert d_i\vert
para todo i = 1, …, n , y que αe ≥ 2 se puede asegurar que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle d_i^2-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\alpha }_e\quad \mu \quad \vert d_i\vert <0
para todo i = 1, …, n , y en consecuencia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\left(\boldsymbol{\mbox{y}}_{\nu }-\right. }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \overline{\boldsymbol{\mbox{y}}}\right)}^t(\boldsymbol{\mbox{y}}_r- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overline{\boldsymbol{\mbox{y}}})<0
. Solo cuando αe = 1 el producto escalar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\left(\boldsymbol{\mbox{y}}_{\nu }-\right. }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \overline{\boldsymbol{\mbox{y}}}\right)}^t(\boldsymbol{\mbox{y}}_r- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overline{\boldsymbol{\mbox{y}}})
puede ser igual a cero, lo cual provocaría el colapso del simplex entero. □
Lema 1.
Sean: un vector y = (y1 , …, yn )t de componentes enteros definido en el espacio euclidiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
, ξ un número entero positivo, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{k}}}}_n=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left(k_1,\ldots ,k_n\right)}^t
un vector cuyas componentes ki ∈ { −1, 0, 1}, para todo i = 1, …, n . Entonces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \xi {\tilde{\boldsymbol{\mbox{k}}}}_n
también pertenece al espacio euclidiano entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
, y su distancia a y es de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \xi \sqrt{n-l}}
, donde l ≤ n es el número de componentes nulas en el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{k}}}}_n}
.
Demostración. En virtud de que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left(y_1,\ldots ,y_n\right)}^t\in {\mathbb{Z}}^n
, entonces,
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( 14) |
Debido al hecho de que cada i -ésima componente de yk indicada en la ecuación (14) , es decir, yi + ξki para todo i = 1, …, n es un número entero, puede asegurarse que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_k\in {\mathbb{Z}}^n}
. Por otra parte, la norma euclidiana del vector definido por la diferencia entre yk e y es
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( 15) |
en virtud de que para todo i = 1, …, n los números escalares ki ∈ { −1, 0, 1}. La ecuación (15) determina la distancia de y a un punto vecino yk . □
Proposición 2 Punto expandido entero.
Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
un q -ésimo simplex entero jerarquizado de acuerdo con f (v ), definido exclusivamente en el espacio euclidiano entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
. Si el punto expandido entero ye es definido por
|
( 16) |
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_e\in \mathbb{N}(>0)}
representa el coeficiente de expansión entero e yr es determinado por la ecuación (8) . Entonces, se cumple que:
- ye es un punto en el espacio de los números enteros Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
;
- con respecto al vértice yν , ye nunca está más cerca de yν , de lo que está yr de yν .
Demostración. Parte I) Represente yr = (y1,r , …, yn ,r )t un punto reflejado producto de la operación definida por la ecuación (8) . Entonces
|
( 17) |
donde ki ∈ { −1, 0, 1} para todo i = 1, …, n debido a la función signo discreta. En consecuencia, cada una de las componentes de ye es entera, debido al hecho de que yi ,r , βe y ki son todos números enteros, para cada i = 1, …, n su operación arroja un número entero.
Parte II) Claramente el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle sgnd\quad (\overline{\boldsymbol{\mbox{y}}}-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \boldsymbol{\mbox{y}}_{\nu })
tiene la misma orientación del vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{y}}}-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \boldsymbol{\mbox{y}}_{\nu }
, y este último va desde el punto yν hasta el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{y}}}}
. Como resultado de esto, el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle sgnd\quad (\overline{\boldsymbol{\mbox{y}}}-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \boldsymbol{\mbox{y}}_{\nu })
tiene como orientación un vector que va desde el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{y}}}}
hasta un punto que se encuentra en la región discreta entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{y}}}}
e yr , y la norma euclidiana de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_e\quad sgnd\quad (\overline{\boldsymbol{\mbox{y}}}-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \boldsymbol{\mbox{y}}_{\nu })
, según el lema 1 , es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_e\quad \sqrt{n-l}\not<0} , lo cual, al sumarse con yr , se ubica en un punto que no está más cerca de lo que está yr de yν .
Proposición 3 Punto contraído entero.
Sea un q -ésimo simplex entero jerarquizado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
de acuerdo a f (v ), definido únicamente en el espacio euclidiano entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
. Si el punto contraído entero yc es determinado por
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( 18) |
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\gamma }_e\in \mathbb{N}(>0)}
es el llamado coeficiente de contracción entero, yr es determinado por (8) . Entonces, se tiene que:
- yc es un punto en el espacio de los números enteros Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
;
- con respecto al vértice yν está más cerca, de lo que está yr de yν .
Demostración. Aplicando el mismo método empleado en la demostración de la proposición 2 , se demuestra la parte I) y parte II) de la proposición.
Proposición 4 Simplex entero encogido.
Sea un q -ésimo simplex entero jerarquizado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
de acuerdo a f (v ), definido solamente en el espacio euclidiano entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^n}
. Si 0 < δe < 1 es el coeficiente de encogimiento entero, y además,
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( 19) |
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{A}}_1^{\left[q\right]}}
es la matriz de arista de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
con respecto al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q\right]}}
, y el operador ⌈· ⌉ aproxima cada elemento de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{A}}_1^{\left[q\right]}}
al próximo entero por encima. Entonces, se cumple que:
- la matriz formada por
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( 20) |
es un simplex compuesto de elementos enteros.
- las aristas del simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{Y}}}}_{\nu }^{\left[q+1\right]}}
con respecto al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q\right]}}
nunca son mayores a las aristas del simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{Y}}_{\nu }^{\left[q\right]}}
con respecto al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q\right]}}
, en consecuencia el nuevo simplex es igual o menor en tamaño.
Demostración. Parte I) Del análisis de la operación definida por la ecuación (19) se puede afirmar que esta arroja como resultado una matriz de dimensión n × n compuesta de elementos enteros, en virtud de que el operador ⌈· ⌉ aproxima cada uno de los elementos de la matrix al próximo entero por encima.
Parte II) Para demostrar esta afirmación, se tiene primero que, de acuerdo con la ecuación (20) , sendas primeras columnas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{S}}}}_{\nu }^{\left[q+1\right]}}
son iguales al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q\right]}}
. Ahora se deberán estudiar las normas de las aristas de sendos símpleces con respecto al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q\right]}}
.
Se tiene que la matriz de aristas del simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
con respecto al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q\right]}}
está definida por
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( 21) |
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{a}}_i^{\left[q\right]}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left(a_{1i},a_{2i},\ldots ,a_{ni}\right)}^t
para todo i = 1, …, n .
Por otra parte, la matriz de arista del simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{S}}}}_{\nu }^{\left[q+1\right]}}
con respecto al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q+1\right]}}
, de acuerdo a la ecuación (20) viene dada por
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( 22) |
Ahora, al comparar la norma euclidiana de cada arista de ambas matrices, es decir, las matrices dadas por las ecuaciones (21) y (22) , se tiene que para todo j = 1, …, n
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( 23) |
Debido al hecho de que 0 < δe < 1, para cada i = 1, …, n y j = 1, …, n , siempre se cumple que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lceil {\delta }_ea_{ij}{\rceil }^2\leq a_{ij}^2}
, lo cual permite asegurar que para todo i = 1, …, n , la norma euclidiana de cada j -ésima arista del simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{S}}}}_{\nu }^{\left[q+1\right]}}
con respecto al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q\right]}}
nunca es mayor a la norma euclidiana para esa misma j -ésima arista del simplex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_{\nu }^{\left[q\right]}}
con respecto al vértice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}_1^{\left[q\right]}}
. □
Observación 4.
La construcción de un nuevo simplex entero encogido mediante la proposición 4 puede producir un simplex entero colapsado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{S}}}}_{\nu }^{\left[q+1\right]}}
. Como ejemplo, nótese que la proposición 4 aplicada al simplex entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_3^{\left[q\right]}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [\boldsymbol{\mbox{y}}_1\quad ;\boldsymbol{\mbox{y}}_2\quad ;\boldsymbol{\mbox{y}}_3]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left[\begin{array}{ccc} 1 & 2 & 3\\ 3 & 1 & 1\\ & & \end{array}\right]
, con una valor de δe = 1/2, arroja como resultado el simplex entero colapsado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{S}}}}_3^{\left[q+1\right]}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left[\begin{array}{ccc} 1 & 2 & 2\\ 3 & 2 & 2\\ & & \end{array}\right]
.
La figura 1 muestra las operaciones de reflexión, expansión, contracción y construcción de un simplex encogido que pueden ser ejecutadas sobre un q -ésimo simplex entero jerarquizado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_3^{\left[q\right]}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [\boldsymbol{\mbox{y}}_1\quad ;\boldsymbol{\mbox{y}}_2\quad ;\boldsymbol{\mbox{y}}_3]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left[\begin{array}{ccc} 1 & 3 & 1\\ 5 & 2 & 1\\ & & \end{array}\right]
, definido en el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^2}
, cuando los coeficientes tomaron los valores αe = 2, y βe = γe = 1, y δe = 1/2.
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Figura 1. Operaciones sobre un simplex entero jerarquizado. |
Obsérvese que el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\boldsymbol{\mbox{y}}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left(2,\frac{7}{2}\right)}^t
claramente no pertenece a los números Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^2}
, sin embargo, el resto de los puntos arrojados por las distintas operaciones sobre el simplex entero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{S}}_3^{\left[q\right]}}
sí pertenecen al espacio de los números Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{Z}}^2}
.
Por último, nótese además que en la figura 1 se muestra el simplex entero encogido, el cual es denotado por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\boldsymbol{\mbox{S}}}}_3^{\left[q+1\right]}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [\boldsymbol{\mbox{y}}_1\quad ;{\overset{\textasciiacute}{\boldsymbol{\mbox{y}}}}_2\quad ;{\overset{\textasciiacute}{\boldsymbol{\mbox{y}}}}_3]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left[\begin{array}{ccc} 1 & 2 & 1\\ 5 & 4 & 3\\ & & \end{array}\right]
, para lo cual se empleó la proposición 4 , la cual podría arrojar un simplex entero colapsado según la observación 4 . No obstante, el algoritmo propuesto cuenta con etapas, las cuales deben iniciar con un nuevo simplex entero mixto.
5. Procedimientos
El ASEM se basa, entre otros pasos, en 2 procedimientos principales, que se presentan en esta sección. Es importante señalar que el superíndice [q ] es en algunas oportunidades suprimido con el propósito de simplificar la notación, sin que pierda su significado.
El procedimiento mostrado en la figura 2 , llamado Procedimiento de Construcción de un Simplex Entero Mixto (CSEM), permite construir un simplex entero mixto basándose, entre otros parámetros, en los valores de Δr y Δe que son definidos inicialmente por el usuario del ASEM a los efectos de tener control de los parámetros del algoritmo. Nótese que en la figura se emplea la notación 02n , la cual representa un vector nulo de dimensión 2n , es decir, un vector de elementos todos nulos.
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Figura 2. Construcción de un Simplex Entero Mixto. |
Otro procedimiento empleado en el ASEM es el Mejoramiento de un Simplex Entero Mixto (MSEM), el cual tiene como propósito redefinir el simplex entero mixto inicial a través de la orientación de sus aristas. Este procedimiento se presenta en la figura 3 con el objetivo de buscar la mejor orientación del simplex construido por el procedimiento CSEM. Los experimentos numéricos realizados por Brea [3] probaron la eficacia del MSEM.
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Figura 3. Mejoramiento de un Simplex Entero Mixto. |
6. El algoritmo
Para iniciar la búsqueda de un mínimo local del problema 1 , el ASEM, mostrado en la figura 4 , arranca con sus parámetros definidos por el usuario, el número de la etapa s = 1 y un punto de inicio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{v}}_0\in {\mathbb{R}}^w\times {\mathbb{Z}}^w}
, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): max(n,m)
. Con el punto de inicio v0 y los parámetros, el algoritmo construye un simplex entero mixto mediante la ejecución del procedimiento CSEM. Si el ASEM es ejecutado bajo la modalidad de selección del mejor simplex entero mixto construido, el cual se indica con m = 1, el algoritmo ejecuta el procedimiento MSEM e inicia el contador de iteraciones q en cero.
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Figura 4. Diagrama del Algoritmo Simplex Entero Mixto. |
Una vez construido un simplex entero mixto inicial, el ASEM evalúa la función objetivo para cada uno de los vértices del simplex actual de los que no se conoce su valor, mediante el paso (3) Evaluación . Seguidamente, se determina el orden de precedencia de sus vértices a través del paso (4) Ordenamiento , lo cual permite determinar el punto de prueba reflejado mediante la ejecución del paso (5) Reflexión . Al final de este último paso, el ASEM selecciona uno de los 4 posibles casos, en función de la precedencia del punto de prueba reflejado, denotado por vr con relación a: v1 , vν −1 y vν . Si vr ≺ v1 , es decir, si f (vr ) < f (v1 ), el ASEM ejecuta una operación de expansión, para luego decidir si el vértice vν será sustituido por el punto de prueba vr o ve . Por otra parte, si v1 ⪯ vr ⪯ vν −1 , el ASEM reemplaza el vértice vν por el punto de prueba vr . No obstante, si vν −1 ≺ vr ≺ vν el algoritmo sustituye el vértice vν del simplex actual por el punto de prueba vr para luego ejecutar el paso (7) Contracción . Finalmente, en el caso de que vν ⪯ vr , el ASEM ejecuta una contracción del simplex actual por medio del paso (7) Contracción .
Después de que el ASEM tome una de las opciones de acuerdo con la precedencia de vr con relación a v1 , vν −1 y vν , el algoritmo determina la cualidad del nuevo simplex de acuerdo al diámetro del simplex X , el cual es denotado por ℓx . Este valor de ℓx es comparado con un criterio de parada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\kappa }_x^{\left[s\right]}}
de la s -ésima etapa, el cual adquiere distintos valores según la etapa que se esté ejecutando. Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {l}_x\geq {\kappa }_x^{\left[s\right]}}
, el ASEM ejecuta una nueva iteración. En caso contrario, el algoritmo detiene las iteraciones dentro de la etapa actual para decidir si continúa la búsqueda o no, de acuerdo con el valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Delta =}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vert \vert {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}}}^{\left[s\right]}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}}}^{\left[s-1\right]}\vert \vert
. Si Δ ≥ ɛ , el ASEM asigna a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\kappa }_x^{\left[s\right]}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \varphi {\kappa }_x^{\left[s-1\right]}
; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Delta }_r^{\left[s\right]}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \rho {\Delta }_r^{\left[s-1\right]}
y con v1 va al paso (2) Construcción de un Simplex Entero Mixto a fin de rearrancar una nueva etapa. En caso contrario, el ASEM reporta a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}}}^{\left[s\right]}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}}}_1
como el mínimo.
7. Flexibilidad del Algoritmo Ximplex Entero Mixto
Una característica particular que tiene el ASEM es su flexibilidad con respecto a problemas en donde el número de componentes reales y enteras son diferentes, es decir, problemas del tipo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^m}
, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\in {\mathbb{N}}_+}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m\in {\mathbb{N}}_+}
pueden ser diferentes.
Para tratar este tipo de problemas se debe convertir el problema de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^m}
a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^{max(n,m)}\times {\mathbb{Z}}^{max(n,m)}}
, para así permitir definir un simplex entero mixto con igual número de componentes reales y enteras.
Es importante destacar que la función objetivo definida en el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^n\times {\mathbb{Z}}^m}
es tratada con su número original de variables. No obstante, el simplex es definido con max(n , m ) + 1 vértices enteros mixtos todos definidos en espacio euclidiano entero mixto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\mathbb{R}}^{max(n,m)}\times {\mathbb{Z}}^{max(n,m)}}
.
Esta generalización de los problemas enteros mixtos se muestra a través de las funciones de pruebas presentadas en el apéndice A .
8. Sintonización del Algoritmo Ximplex Entero Mixto
Para el proceso de sintonización del ASEM se consideraron los parámetros obtenidos en la sintonización del Algoritmo Simplex Entero Relajado (ASER) desarrollado por Brea [3] , además de los valores recomendados y apoyados en las investigaciones de Brea sobre el MSNM cuando el problema de optimización no lineal está bajo restricciones lineales [5] (tabla 1 ).
En consecuencia, los parámetros αr , βr , γr y δr indicados en la tabla 1 fueron ajustados de acuerdo a los valores recomendados en Nelder y Mead [2] , mientras los parámetros Δe , αe , βe , γe y δe se fijaron de acuerdo con los valores obtenidos en el proceso de sintonización del ASER (véase Brea [3, cap. 5] ). Además, los valores de los parámetros φ y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_x^{\left[1\right]}}
mostrados en la tabla 1 se obtuvieron mediante pruebas preliminares, las cuales permitieron mostrar un desempeño satisfactorio del ASEM. Finalmente, el parámetro ɛ representa el criterio de comparación de los resultados obtenidos en 2 etapas sucesivas y, en consecuencia, debe ser fijado por el usuario dependiendo del máximo error que espera obtener en la comparación de 2 etapas sucesivas.
| Parámetro | ɛ | φ | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_x^{\left[1\right]}} | ||
| Valores | 0,1 | 0,3 | 1 | ||
| Parámetro | αr | βr | γr | δr | |
| Valores | 1 | 2 | 0,5 | 0,5 | |
| Parámetro | Δe | αe | βe | γe | δe |
| Valores | 1 | 2 | 2 | 1 | 0,4 |
Estas consideraciones permitieron simplificar significativamente la experimentación para la obtención de los valores apropiados de ρ y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Delta }_r^{\left[1\right]}}
. Es importante señalar que durante la realización de los experimentos, el número máximo de iteraciones que se le permitió ejecutar al ASEM por etapa fue de 15.000, debido a que el ASEM podía quedar en un proceso iterativo, sin encontrar un avance significativo en la minimización de la función objetivo.
La tabla 2 muestra el número de evaluaciones (NE) ejecutadas sobre la función objetivo FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}}
[d + d ] con ai = bi = 1 para todo i = 1, …, d , de acuerdo con los valores de ρ y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Delta }_r^{\left[1\right]}}
indicados en la tabla.
| d = 2 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho \setminus {\Delta }_r^{\left[1\right]}} | 0,5 | 1,0 | 1,5 |
|---|---|---|---|---|
| 0,7 | 175 (288e-6) | 127 (163e-6) | 133 (169e-6) | |
| 0,8 | 192 (10e-6) | 115 (14e-6) | 146 (19e-6) | |
| 0,9 | 107 (214e-6) | 165 (71e-6) | 169 (2) | |
| d = 4 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho \setminus {\Delta }_r^{\left[1\right]}} | 0,5 | 1,0 | 1,5 |
| 0,7 | 529 (5e-6) | 457 (1) | 425 (100e-6) | |
| 0,8 | 727 (2e-6) | 435 (174e-6) | 544(107e-6) | |
| 0,9 | 883 (15e-6) | 379 (174e-6) | 544 (3) | |
| d = 8 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho \setminus {\Delta }_r^{\left[1\right]}} | 0,5 | 1,0 | 1,5 |
| 0,7 | 12498 (0) | 3123 (1e-6) | 5452 (3e-6) | |
| 0,8 | 14192 (0) | 7651 (0) | 4835 (2e-6) | |
| 0,9 | 5686 (0) | 2761 (2) | 3642 (1) | |
| d = 14 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho \setminus {\Delta }_r^{\left[1\right]}} | 0,5 | 1,0 | 1,5 |
| 0,7 | 57251 (90) | 96558 (0) | 45253 (1) | |
| 0,8 | 17107 (1e-6) | 80936 (0) | 38234 (0) | |
| 0,9 | 5686 (0) | 2761 (2) | 3642 (1) | |
| d = 18 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho \setminus {\Delta }_r^{\left[1\right]}} | 0,5 | 1,0 | 1,5 |
| 0,7 | 35378 (0) | 270236 (596e-6) | 40550 (3) | |
| 0,8 | 188136 (0) | 72880 (0) | 58245 (2) | |
| 0,9 | 83971 (0) | 146017 (3e-6) | 33041 (5) | |
| d = 20 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho \setminus {\Delta }_r^{\left[1\right]}} | 0,5 | 1,0 | 1,5 |
| 0,7 | 405221 (3952e-6) | 253326 (1009) | 71239 (2) | |
| 0,8 | 281525 (0) | 161640 (0) | 57712 (1) | |
| 0,9 | 95165 (0) | 160039 (0) | 68075 (4) |
Además, en la tabla 2 se muestra entre paréntesis el mejor valor de la función objetivo para el NE indicado, siendo en todos los casos el punto óptimo en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{\boldsymbol{\mbox{x}}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \boldsymbol{0}_d
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{\boldsymbol{\mbox{y}}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \boldsymbol{0}_d
, con un valor de la función objetivo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}})=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0
.
Como ejemplo de lectura de la tabla 2 se tiene que para d = 8, ρ = 0, 8 y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Delta }_r^{\left[1\right]}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1
el NE fue igual a 7.651 y el valor mínimo obtenido de la función objetivo fue entre 0 y un valor menor a 10−6 , en particular fue igual a 7.67546044785799 × 10−8 .
De la tabla 2 se puede apreciar que unos valores adecuados para ρ y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Delta }_r^{\left[1\right]}}
son 0,8 y 1 respectivamente, debido al hecho de que para esos valores el ASEM identificó el punto mínimo de la función objetivo para todos los casos de dimensión entera mixta considerados.
Es importante destacar que estos valores de sintonización del ASEM son referenciales y que no deben tomarse como valores que deben satisfacer la totalidad de los casos de estudio, puesto que fueron obtenidos de casos particulares.
9. Experimentos numéricos
La tabla 3 muestra los valores mínimos obtenidos por el ASEM, el mínimo teórico y el orden Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}
de magnitud del error e , definido como
|
( 24) |
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{f}(\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}})}
representa el valor mínimo determinado por el ASEM Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}})}
representa el valor mínimo teórico para cada una de las funciones de prueba definidas en el Apéndice A .
| Problema | Condiciones | Mínimo | Total | |||||
|---|---|---|---|---|---|---|---|---|
| ρ | x0 | y0 | ASEM | Teórico | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O(e)} | NE | NI | |
| FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[5+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] |
0, 85 | 10 · 15 | 10 · 110 | 3, 85399595970641 × 10−17 | 0 | 10−16 | 2.687 | 1.525 |
| FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[10+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 5] |
0, 85 | 10 · 110 | 10 · 15 | 3, 19278384117684 × 10−7 | 0 | 10−6 | 22.718 | 14.854 |
| FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[10+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] |
0, 85 | 10 · 110 | 10 · 110 | 3, 2938056148257 × 10−11 | 0 | 10−10 | 27.550 | 18.956 |
| FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[20+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 20] |
0, 85 | 10 · 120 | 10 · 120 | 1, 42023676961727 × 10−9 | 0 | 10−8 | 156.785 | 121.154 |
| FCDFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[5+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] |
0, 85 | 10 · 15 | 10 · 110 | 0,777777777837533 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{7}{9}} | 10−9 | 15.428 | 10.553 |
| FCDFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[10+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 5] |
0, 85 | 10 · 110 | 10 · 15 | 0,444444465663501 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{4}{9}} | 10−7 | 5.511 | 3.551 |
| FCDFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[10+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] |
0, 85 | 10 · 110 | 10 · 110 | 0,777777777837533 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{7}{9}} | 10−9 | 15.428 | 10.553 |
| FCDFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[20+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 20] |
0, 85 | 10 · 120 | 10 · 120 | 1, 55555555557834 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{14}{9}} | 10−10 | 68.884 | 53.399 |
| FGMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[5+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] |
0, 85 | 10 · 15 | 10 · 110 | 3, 1137516964225 | - | - | 1.639 | 863 |
| FGMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[10+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 5] |
0, 85 | 10 · 110 | 10 · 15 | −0, 9998056161237 | - | - | 3.036 | 1.942 |
| FGMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[10+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] |
0, 85 | 10 · 110 | 10 · 110 | 3, 91853917817428 | - | - | 2.982 | 1.703 |
| FGMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[20+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 20] |
0, 85 | 10 · 120 | 10 · 120 | 0, 995491554751617 | - | - | 55.667 | 43.607 |
| FPMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[5+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 5] |
0, 85 | 10 · 15 | 10 · 15 | 1, 91104307081247 × 10−5 | 0 | 10−4 | 1.276 | 371 |
| FPMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[10+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] |
0, 85 | 10 · 110 | 10 · 110 | 0, 00596898501698284 | 0 | 10−2 | 2.172 | 927 |
| FPMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[20+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 20] |
0, 85 | 10 · 120 | 10 · 120 | 3, 07 × 10−5 | 0 | 10−4 | 27.897 | 17.457 |
| FRFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[10+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] |
0, 99 | 5 · 110 | 5 · 110 | 6, 1531 × 10−13 | 0 | 10−12 | 10.213 | 6.677 |
| FRFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[20+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 20] |
0, 99 | 5 · 120 | 5 · 120 | 4, 75 × 10−6 | 0 | 10−5 | 48.777 | 36.038 |
| FADFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[2+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1] |
0, 85 | 10 · 12 | 10 · 11 | −13, 9942 | -14 | 10−2 | 272 | 67 |
Adicionalmente, la tabla 3 presenta el número de evaluaciones (NE) de la función objetivo y el número de iteraciones (NI) totales ejecutadas por el ASEM, para las funciones de prueba. La tabla también suministra las condiciones con que fue ejecutado el ASEM, tales como el factor ρ que significa la tasa de reducción del paso de cada simplex de arranque por etapa, así como el punto inicial v0 , donde 1d es el vector unitario de dimensión d . Además, el resto de los parámetros fueron ajustados de acuerdo con los valores indicados en la tabla 1 .
Nótese que para los tipos de problemas denotados como FGMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[n+} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): m]
no se especificó el valor teórico debido a que esos tipos de problemas presentan múltiples mínimos locales y se ha supuesto que el mínimo identificado por el ASEM es un mínimo local al problema, mas no el mínimo global.
Para los casos en donde fue considerado un punto de inicio v0 con componentes todas iguales a 10, se puede inferir que, de haberse identificado sus respectivos mínimos por búsqueda exhaustiva, solo por las variables enteras, se hubiesen requerido 1020 evaluaciones, es decir, 100 × 106 T evaluaciones de la función objetivo, lo cual permite inferir la alta eficiencia que posee el ASEM desde el punto de vista del NE, en el caso de funciones objetivo convexas.
La figura 5 muestra la gráfica de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\boldsymbol{\mbox{v}}_1^{\left[q\right]})}
del problema FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[5+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10]
en función de la iteración q del ASEM, para un valor de ρ = 0, 8, dejando ajustado el resto de los parámetros según lo indicado en la tabla 1 , y punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{v}}_0^t=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (10\cdot \boldsymbol{1}_5^t,10\cdot \boldsymbol{1}_{10}^t)
. La figura muestra al final de cada s -ésima etapa, sobre la escala denotada por q , el valor alcanzado del número de iteraciones q . Por otro lado, en la parte inferior de la figura se muestra la escala correspondiente al número de la etapa s del ASEM, y es denotada por s . Nótese además que en la figura se representa el valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\boldsymbol{\mbox{v}}_1^{\left[1\right]})}
, es decir, al final de la primera iteración, cuyo valor fue de 1462.
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![Evolución de la función objetivo para el Problema FCℝℤ[5+10], para el caso ...](/wd/index.php/File:Draft_Garc%C3%ADa_644613932-1-s2.0-S0213131513000345-gr5.jpg)
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Figura 5. Evolución de la función objetivo para el Problema FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[5+} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10] , para el caso cuando ρ = 0, 8. |
Por otra parte, la tabla 4 presenta un resumen por etapa del ASEM, cuando este fue aplicado al problema FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[5+} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 10]
bajo las mismas condiciones con que fue reportada la figura 5 . Nótese que los datos numéricos que se muestran en la columna q corresponden al número máximo de iteraciones empleadas por el ASEM en cada etapa, así como el NE de la función objetivo.
| s | ℓx | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\kappa }_x^{\left[s\right]}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Delta }_r^{\left[s\right]}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f({\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}}}^{\left[s\right]})} | q | NE |
|---|---|---|---|---|---|---|
| 1 | 0,55972 | 1 | 1 | 397,8225 | 44 | 109 |
| 2 | 0,25699 | 0,3 | 0,8 | 27,1793 | 385 | 671 |
| 3 | 0,07269 | 0,09 | 0,064 | 9,0016 | 124 | 250 |
| 4 | 0,01749 | 0,027 | 0,512 | 2,488×10−6 | 170 | 334 |
| 5 | 0,00766 | 0,0081 | 0,4096 | 7,379×10−13 | 267 | 491 |
10. Comparación con otros métodos
El problema del diseño óptimo de un tanque presurizado ha servido de ejemplo de aplicación y comparación de diversos métodos algorítmicos de optimización, entre los cuales se pueden señalar los métodos de optimización estudiados por Zahara y Kao [8] y por Tahera et al. [9] .
La figura 6 muestra la concha semiesférica izquierda y el cilindro de un tanque presurizado, mas no la concha semiesférica derecha que es también parte de la estructura del tanque. La gráfica presenta la geometría del tanque que hay que diseñar junto a las variables de decisión: R (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_1\in \mathbb{R}}
, radio interno del cilindro y de las conchas semiesféricas medido en pulgadas); L (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_2\in \mathbb{R}}
, longitud del cilindro en pulgadas); Ep (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y_1\in \mathbb{Z}}
, espesor de la pared del cilindro en cantidades enteras de la fracción Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{16}}
pulgadas, debido a los distintos calibres con que es fabricado el material) y Ec (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y_2\in \mathbb{Z}}
, espesor de la pared de las conchas semiesféricas en cantidades enteras de la fracción Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{16}}
pulgadas).
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Figura 6. Tanque cilíndrico presurizado. |
El problema en términos matemáticos presentado por Zahara y Kao [8] and [9] , que expresa en dólares americanos el costo de producir un tanque cilíndrico presurizado en el que se adecuan las variables a la notación empleada en esta investigación, viene dada por
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donde para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{x}}\in {\mathbb{R}}^2}
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{y}}\in {\mathbb{Z}}^2}
,
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( 25) |
sujeto a:
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( 26) |
Con el propósito de redefinir el problema como un problema de minimización irrestricto, se definieron las siguientes funciones de penalización:
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( 27) |
lo que permitió redefinir el problema a través de las funciones de penalización dadas por la ecuación (27) como:
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( 28) |
Para la identificación del mínimo se empleó el ASEM con sus parámetros ajustados de acuerdo con lo especificado por la tabla 1 y con valores de ρ = 0, 8 y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Delta }_r^{\left[1\right]}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1
, y como punto de inicio se fijaron todas las componentes en 10.
El resultado obtenido a través del ASEM se comparó con los resultados de otros métodos tales como: el GADYM, acrónimo del inglés Gender-Age structure DYnamic parameter tuning and Mandatory self perfection scheme , propuesto por Tahera et al. [9] ; el GRTA, del inglés Generalized Random Tunneling Algorithm , desarrollado por Kitayama y Yamazaki [10] ; el NM-PSO, acrónimo del inglés Nelder-Mead simplex search method and Particle Swarm Optimization , de Zahara y Kao [8] y el GRG, conocido del inglés Generalised Reduced Gradient , reportado también en Tahera et al. [9] , los cuales son indicados en la tabla 5 .
| Variables medidas en pulgadas [in] | ||||||
|---|---|---|---|---|---|---|
| Algoritmo | f (x , y ) | S (L , R ) [in2 ] | R (x1 ) | L (x2 ) | Ep (y1 ) | Ec (y2 ) |
| GADYM | 6.062,65 | 69.026,731 | 42,098 | 176,769 | 0,4357 | 0,8125 |
| GRTA | 6.059,58 | 68.992,022 | 42,098 | 176,634 | 0,4375 | 0,8125 |
| NM-PSO | 5.930,31 | 69.511,951 | 41,639 | 182,412 | 0,3972 | 0,8036 |
| GRG | 9.664,09 | 74.691,075 | 37,692 | 240,000 | 1 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{16}}
) |
1 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{16}}
) |
| ASEM | 8.796,92 | 60.623,882 | 51,807 | 84,627 | 1 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{16}}
) |
1 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{16}}
) |
La tabla 5 muestra los valores obtenidos de la función objetivo y sus correspondientes valores de las variables de decisión para el mejor punto identificado por cada algoritmo. Además, incluye el valor de la superficie, S (L , R ) = 2πRL + 4πR2 , de material requerido para la construcción del envase presurizado, puesto que la comparación debe hacerse según la superficie de material requerido, en virtud de que únicamente el GRG y el ASEM tomaron en cuenta la naturaleza entera de las variables Ec y Ep , las cuales están indicadas en la tabla mediante el número de fracciones de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{16}}
pulgadas. El hecho de no poder asignar valores reales a las variables Ec y Ep produce un aumento en la función objetivo con relación a los valores que hubieran tomado Ec y Ep , en caso de haberse considerado variables reales. Nótese que de acuerdo con los resultados mostrados en la tabla 5 , el ASEM alcanzó a identificar la menor superficie de material para la fabricación del envase presurizado, con sólo 440 NE, mientras que el resto de los algoritmos emplearon decenas de miles de evaluaciones de la función objetivo.
Más aún, el ASEM identificó un mejor resultado en comparación al algoritmo GRG, el cual es un algoritmo para problemas de OEM.
11. Conclusiones
Como conclusiones de este trabajo de investigación se tiene que:
- El empleo de la estructura del simplex dentro del campo de los números enteros, junto a las redefinidas operaciones de reflexión, extensión, contracción y encogimiento, result’0 muy apropiado para definir una estrategia de identificación de óptimos de problemas enteros mixtos irrestrictos, sin el relajamiento de las variables enteras de decisión.
- La principal característica que tienen los problemas de variables enteras mixtas en cuanto a la derivabilidad de sus funciones objetivo es que estas no cuentan con expresiones de derivadas con respecto a las variables enteras, desde el punto de vista estricto de la definición de derivada. Este hecho ha llevado a desarrollar un método libre de derivadas, además de haber desarrollado métodos algorítmicos que requieren un relativo bajo número de evaluaciones de la función objetivo.
Por otra parte, el desarrollo de otros enfoques para la identificación de óptimos así como de estructuras teóricas son temas de investigación que podrían constituir interesantes investigaciones, debido al reto que imponen. En específico, se pueden sugerir como temas de investigación:
- El estudio teórico de las condiciones de optimalidad dentro de la programación no lineal entera mixta.
- El desarrollo de funciones de penalización para el empleo del ASEM con problemas bajo restricciones. Este último puede constituir un importante trabajo de investigación, debido a las innumerables aplicaciones que tiene este tipo de problemas dentro de la ingeniería.
Agradecimientos
El autor desea expresar su gratitud al Consejo de Desarrollo Científico y Humanístico de la Universidad Central de Venezuela por la financiación de esta investigación a través del proyecto PI-08-7506-2009/1. Él desea realmente agradecer también a los árbitros por sus valiosos comentarios y sugerencias, los cuales permitieron mejorar el artículo.
Appendix A. Funciones de prueba
En este apartado son mostrados los problemas enteros mixtos empleados en los experimentos numéricos, para lo cual se empleó una notación que permite definir la dimensión del problema de optimización. Además, para cada uno de los problemas mostrados se establece su respectiva solución óptima.
Función cuadrática entera mixta: FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}}
[n + m ]Función cuadrática entera mixta: FCFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}}
[n + m ]
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( 29) |
donde d es la dimensión del problema, y ai y bi para i = 1, …, d representan coeficientes reales positivos.
Solución. El mínimo global está ubicado en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\overset{\mbox{ˆ}}{\mbox{x}}}}^t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (\underset{n}{\underbrace{0,0,\ldots ,0}})
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\overset{\mbox{ˆ}}{\mbox{y}}}}^t=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (\underset{m}{\underbrace{0,0,\ldots ,0}})
con un valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}})=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0
.
Función cuadrática entera mixta desplazada del origen: FCDFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}}
[n + m ]Función cuadrática entera mixta desplazada del origen: FCDFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}}
[n + m ]
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( 30) |
donde n es la dimensión del subvector real, m es la dimensión del subvector entero, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a_i\in \mathbb{R}}
para i = 1, …, n , y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b_i\in \mathbb{R}}
para i = 1, …, m representan coeficientes reales positivos, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_{0i}\in \mathbb{R}}
para i = 1, …, n , y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y_{0i}\in \mathbb{R}}
para i = 1, …, m representan las coordenadas del vértice de la función cuadrática.
Para el caso particular de los experimentos numéricos ejecutados, se consideró ai = 1 y x0i = i /3 para todo i = 1, …, n , bi = 1 y y0i = i /3 para todo i = 1, …, m .
Solución. El mínimo global está ubicado en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\overset{\mbox{ˆ}}{\mbox{x}}}}^t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (\underset{n}{\underbrace{x_{01},x_{02},\ldots ,x_{0n}}})
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\overset{\mbox{ˆ}}{\mbox{y}}}}^t=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (\underset{m}{\underbrace{{\overset{\mbox{ˆ}}{y}}_1,{\overset{\mbox{ˆ}}{y}}_2,\ldots ,{\overset{\mbox{ˆ}}{y}}_m}})
, donde
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( 31) |
con un valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}})=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\sum }_{i=1}^mb_i{\left({\overset{\mbox{ˆ}}{y}}_i-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. y_{0i}\right)}^2
.
Función producto de módulos entero mixto: FPMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[d+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): d]
Función producto de módulos entero mixto: FPMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[d+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): d]
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( 32) |
donde d es la dimensión del problema y ai para todo i = 1, …, d representan coeficientes reales positivos.
Solución. El mínimo global está ubicado en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{x}}}}^t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (\underset{d}{\underbrace{0,0,\ldots ,0}})
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{y}}}}^t=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (\underset{d}{\underbrace{0,0,\ldots ,0}})
con un valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}})=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0
.
A.1. Función Rosenbrock: FRFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[d+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): d]
A.1. Función Rosenbrock: FRFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[d+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): d]
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donde d es la dimensión del problema y es un número par positivo.
Solución. El mínimo global está ubicado en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{x}}}}^t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(\underset{d}{\underbrace{1,1,\ldots ,1}}\right)
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{y}}}}^t=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(\underset{d}{\underbrace{1,1,\ldots ,1}}\right)
con un valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\overset{\mbox{ˆ}}{\boldsymbol{\mbox{y}}})=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0
.
Función Audet & Dennis: FADFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[2+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1]
Función Audet & Dennis: FADFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[2+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1]
El problema original de [11] es:
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sujeto a:
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donde
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y las funciones g (x1 , x2 ) y h (x1 , x2 ) para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol{\mbox{x}}\in {\mathbb{R}}^2}
vienen dadas por
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El planteamiento del problema, empleando el concepto de función de penalización, queda entonces de la siguiente forma:
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donde para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_1\in \mathbb{R}}
; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_2\in \mathbb{R}}
; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y_1\in \mathbb{Z}}
y k = 103
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Solución. El mínimo global está ubicado en el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{x}}}}^t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (-2,-2)
e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{\boldsymbol{\mbox{y}}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (0)
con un valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(\overset{\mbox{ˆ}}{\boldsymbol{\mbox{v}}})=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -14
.
Función Griewank Modificada: FGMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[n+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): m]
Función Griewank Modificada: FGMFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbb{R}\mathbb{Z}[n+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): m]
Una versión modificada en el campo de los números enteros mixtos al problema de optimización de la función de Griewank presentada por Fu et al. [12] se muestra a continuación:
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References
- [1] R. Hooke, T.A. Jeeves; “Direct Search” Solution of Numerical and Statistical Problems; Journal of the Association for Computing Machinery, 8 (2) (1961), pp. 212–229
- [2] J.A. Nelder, R. Mead; A simplex method for function minimization; The Computer Journal, 7 (1965), pp. 308–313
- [3] E. Brea, (2009) Extensiones del método de Nelder Mead a problemas de variables enteras y enteras mixtas. Trabajo de Ascenso a la categoría de Profesor Titular, Universidad Central de Venezuela. Caracas, Venezuela.
- [4] J. Brandts, S. Korotov, M. Křížek, J. Šolc; On Nonobtuse Simplicial Partitions; SIAM Review, 51 (2) (2009), pp. 317–335
- [5] E. Brea, (2004) Nelder-Mead Optimization Under Linear Constraints. Ph.D. Thesis, University of Southampton, Southampton, England.
- [6] R.R. Barton, J.S. Ivey Jr.; Nelder-Mead simplex modifications simulation optimization; Management Science, 42 (7) (1996), pp. 954–973
- [7] D.G. Humphrey, J.R. Wilson; A revised simplex procedure for stochastic simulation response surface optimization; INFORMS Journal on Computing, 12 (4) (2000), pp. 272–283
- [8] E. Zahara, Y.T. Kao; Hybrid Nelder-Mead simplex search and particle swarm optimization for constrained engineering design problems; Expert Systems with Applications, 36 (2, Part 2) (2009), pp. 3880–3886
- [9] K. Tahera, R.N. Ibrahim, P.B. Lochert; CADYM - A novel genetic algorithm in mechanical design problems; Journal of Universal Computer Science, 14 (15) (2008), pp. 2566–2581
- [10] S. Kitayama, K. Yamazaki; Global Optimization by Generalized Random Tunneling Algorithm (4th Report Application to the Nonlinear Optimum Design Problem of the Mixed Design Variables); Journal of Computational Science and Technology, 2 (1) (2008), pp. 258–267
- [11] C. Audet, J.E. Dennis Jr.; Pattern Search Algorithms for Mixed Variable Programming; SIAM Journal on Optimization, 11 (3) (2001), pp. 573–594
- [12] M. Fu, S. Marcus, J. Hu; Model-Based Randomized Methods for Global Optimization; 17th International Symposium on Mathematical Theory of Networks and Systems, vol. 1, Kyoto, Japan (2006), pp. 355–363
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Published on 30/03/17
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